Clearly, R^2 always lies between zero and one. If the regression equation pre-
dicted the data perfectly (i.e., the predicted and actual values coincided), then
SSE 0, implying that R^2 1. Conversely, if the equation explains nothing
(i.e., the individual explanatory variables did not affect the dependent vari-
able), SSE would equal TSS, implying that R^2 0. In our case, the regression
equation explains 78 percent of the total variation.
Although R^2 is a simple and convenient measure of goodness of fit, it suf-
fers from certain limitations. The value of R^2 is sensitive to the number of
explanatory variables in the regression equation. Adding more variables results
in a lower (or, at least, no higher) SSE, with the result that R^2 increases. Thus,
it is a mistake to regard the main goal of regression analysis as finding the equa-
tion with the highest R^2. We can always jack up the R^2 by throwing more vari-
ables into the right-hand side of the regression equation—hardly a procedure
to be recommended.ADJUSTED R-SQUARED A partial remedy for this problem is to adjust R-squared
according to the number of degrees of freedom in the regression. The number
of degrees of freedom is the number of observations (N) minus the number of esti-
mated coefficients (k). In the airline regression, the number of observations is
16, and the number of coefficients (including the constant term) is 4. Thus, the
degrees of freedom are N k 16 4 12. The adjusted R-squared is given byRˆ^2 1 [4.6]
SSE/(Nk)
TSS/(N1)142 Chapter 4 Estimating and Forecasting DemandTABLE 4.6
Airline Demand
Regression OutputRegression Output
Dependent variable: Q
Sum of squared errors 2616.40
Standard error of the regression 14.77
R-squared 0.78
Adjusted R-squared 0.72
F-statistic 13.9
Number of observations 16
Degrees of freedom 12Constant P P Y
Coefficients 28.84 2.12 1.03 3.09
Standard error of coefficients 0.34 .47 1.00
t-statistic 6.24 2.20 3.09c04EstimatingandForecastingDemand.qxd 9/5/11 5:49 PM Page 142